Why is the particle γὰρ used here? I understand that she blushed, because she mentioned marriage to his father, but why is this fact a cause of the previous clause "ὥς ἔφατο"?I mean, I would understand: "she stopped speaking, for she blushed...", but it says "she spoke, for she blushed...". Is there something wrong with my lecture?

Thanks Paul, now it has much sense. I would like to observe that it also explains the imperfect of αἰδέομαι: I was reading "she blushed, she shamed", as a punctual fact, but I should read "she was ashamed", as a durative action.

Just one more question, which tense of the infinitive would I have to use if I wanted to say "she was ashamed for having mentioned marriage to his father"? Perfect maybe? Or should I leave the infinitive as it is and change the main verb αἰδέομαι to the aorist?

I think the problem is that αἰδέομαι is used in Homeric Greek for something that you don't want to do out of shame or respect, it's not used if you are ashamed for something you have already done. See Cunliffe: